operators on Hilbert spaces have adjoints

Question

The following is a Theorem of Murphy's C*-algebras and operator theory:

In the last line of proof, he claims $u^*$ is linear, but I think it's conjugate linear because for $y_2,x_2\in H_2$, $x_1\in H_1$ and $\alpha \in \Bbb C$ we have $$[u^*(\alpha x_2+y_2)](x_1) = (u x_1, \alpha x_2+ y_2)=[\bar \alpha u^*(x_2)+u^*(y_2)](x_1)$$ Where is my mistake? Please help me. Thanks so much.

Answer 1

$(x,u^*\alpha y)=(ux,\alpha y)=\bar\alpha(x,u^*y)=(x,\alpha u^*y)$.